Question

A ball is projected vertically upwards with speed 21 m/s from a point A, which is 1.5 m above the ground. After projection, the ball moves freely under gravity until it reaches the ground. Modeling the ball as a particle, find
(a) the greatest height above A reached by the ball (found it which is equal to 22.5)
(b) the speed of the ball as it reaches the ground (they used the equation v^2=u^2+2ax and took 'u' zero, but it's not zero? Got a bit confused here..)
(c) the time between the instant when the ball is projected from A and the instant when the ball reaches the ground (and here I just

Answer 1

The acceleration of the particle is a constant \(9.81\dfrac{\text{m}}{\text{s}^2}\). Integrating with respect to time will give you the general velocity function:
\[\begin{align*}
a(t)&=-9.81\\\\
v(t)&=\int a(t)\,dt\\
&=-9.81\int \,dt\\
&=-9.81t+C
\end{align*}\]
We can find the precise value of \(C\) thanks to the initial condition that the velocity at time \(t=0\) is \(21\dfrac{\text{m}}{\text{s}}\):
\[\begin{align*}
v(0)&=-9.81(0)+C\\
21&=0+C\\
C&=21
\end{align*}\]
So, \(v(t)=-9.81t+21\). Integrating again will yield the position function:
\[\begin{align*}
v(t)&=-9.81t+21\\\\
s(t)&=\int v(t)\,dt\\
&=-9.81\int t\,dt+21\int\,dt\\
&=-4.9t^2+21t+C
\end{align*}\]
Again, we can an initial condition to find this \(C\):
\[\begin{align*}
s(0)&=-4.9(0)^2+21(0)+C\\
1.5&=C
\end{align*}\]
So, your position function is \(s(t)=-4.9t^2+21t+1.5\).

The greatest height is achieved when the particle stops moving in mid-air, i.e. when its velocity is \(0\).
\[-9.81t+21=0~~\implies~~t\approx2.14\]
Evaluate the position function for this \(t\):
\[s(2.14)\approx 23.97\text{ m}\]
which doesn't exactly match what you got for (a). Any ideas where either of us might have gone wrong?

Answer 2

For part (b), you would find the value of \(t\) that gives \(s(t)=0\), then evaluate \(v(t)\) for that point. You'll have two possible solutions, one of which you can throw out right away.

For (c), the works is already done in part (b).

Now that I look back at your question, I get the feeling this is a physics problem geared towards non-calculus students...